1975 Canadian MO Problems/Problem 4
Problem 4
For a positive number such as , is referred to as the integral part of the number and as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
Solution
Let be the integer part and be the decimal part, thus, we have the G.P.
So,
There we have a quadratic equation. We must isolate or .
If the number must be positive, thus we'll consider only the solution
As is the decimal part, then it must be lower than 1
This is the golden ratio and it's approximately .
As must be an integer, thus . Therefore
To find our number we must sum , so
The number we're searching for it's the golden ratio.
1975 Canadian MO (Problems) | ||
Preceded by Problem 3 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | Followed by Problem 5 |